= Solution
{c}
It is useful first to establish the global minimum, rather than only a stationary point. The <positive semidefinite matrix> $X=\Phi^\dagger\Phi$ has nonnegative <eigenvalues> $x_1,\ldots,x_N$. For fixed $r=\sum_i x_i$, the <Cauchy-Schwarz inequality> gives $\sum_i x_i^2\ge r^2/N$, with equality exactly when all $x_i$ are equal. Since $\alpha>0$,
$$
V\ge\frac12\left(\frac\alpha N+\beta\right)r^2+m^2r.
$$
For $m^2<0$ and $\beta>0$, this lower bound has its minimum at
$$
r=-\frac{Nm^2}{\alpha+N\beta},\qquad x_i=C^2=-\frac{m^2}{\alpha+N\beta}>0.
$$
The bound is attained, so every minimum has $\Phi^\dagger\Phi=C^2\mathbf1$, or $\Phi=CW$ with $W$ a <unitary matrix>. A global transformation removes $W$ by decomposing it into an overall phase and an $SU(N)$ matrix. Thus $\Phi_0=C\mathbf1$ is a valid representative of the <vacuum manifold>.
Preservation of this chosen <vacuum expectation value> requires
$$
e^{i\theta}U_LU_R^\dagger=\mathbf1.
$$
Infinitesimally, with traceless Hermitian generators $L,R$, this reads $\theta\mathbf1+L-R=0$. Taking the <trace> forces $\theta=0$, and then $L=R$. The surviving continuous transformations are consequently
$$
\boxed{\Phi\mapsto U\Phi U^\dagger,\qquad U\in SU(N),\qquad H_{\mathrm{connected}}\simeq SU(N)_{\mathrm{diagonal}}.}
$$
All other independent continuous directions change the chosen <vacuum expectation value>. Thus the number of broken generators is
$$
\boxed{[2(N^2-1)+1]-(N^2-1)=N^2.}
$$
Globally, taking the determinant of the stabilizer equation permits the discrete phases $e^{iN\theta}=1$, paired with $U_L=e^{-i\theta}U_R$. They induce the same conjugation action on $\Phi$. The faithful unbroken conjugation group is $SU(N)/\mathbb Z_N$; for example, at $N=2$, $U=-\mathbf1$ acts exactly like $U=\mathbf1$. Thus the question's description by $SU(N)$ is correct as a parametrization and at the <Lie algebra> level, with this central redundancy understood. It leaves the number of broken generators unchanged.
Back to article page
